Power allocation for superposition transmission

ABSTRACT

Apparatuses, systems, and methods are described for power allocation in a superposition multiple access communication system capable of using non-uniform joint constellations or super-constellations. In one method, the conditional probability of a correctly-received symbol and a normalized weighting coefficient is calculated for each receiver and then the sum of weighted efficiencies is calculated. The optimal power allocation is determined for each receiver by maximizing the sum of weighted spectral efficiencies.

PRIORITY

This application claims priority under 35 U.S.C. § 119(e) to a U.S.Provisional Patent Application filed on Jun. 9, 2015 in the UnitedStates Patent and Trademark Office and assigned Ser. No. 62/173,241, aU.S. Provisional Patent Application filed on Aug. 11, 2015 in the UnitedStates Patent and Trademark Office and assigned Ser. No. 62/203,818, aU.S. Provisional Patent Application filed on Aug. 12, 2015 in the UnitedStates Patent and Trademark Office and assigned Ser. No. 62/204,305, anda U.S. Provisional Patent Application filed on Aug. 26, 2015 in theUnited States Patent and Trademark Office and assigned Ser. No.62/210,326, the entire contents of each of which are incorporated hereinby reference.

FIELD OF THE DISCLOSURE

The present disclosure relates generally to power management formultiple access communication technologies, and more particularly, topower allocation when performing superposition multiple accesstransmission.

BACKGROUND

The amount of data traffic on wireless communications continues toincrease at an almost exponential rate. For example, many cell phoneusers expect their cell phones to routinely handle both the ability tosurf the Internet at any time and to stream movies-sometimes at the sametime. Thus, new ways of further maximizing data throughput arecontinually discussed and often implemented in each new version of astandard.

One way to increase throughput (when there are multiple receivers) issuperposition multiple access (it is also known by other names), whichwill be described more fully below. This multiple access method hasrecently increased its importance as it is under serious considerationby the standards organization 3^(rd) Generation Partnership Project(3GPP) to be part of the next Long Term Evolution (LTE) release. See,e.g., Chairman's Notes, 3GPP RAN1 Meeting #80b, Belgrade (2014, Apr.20). Within and without 3GPP, the particular implementation ofsuperposition multiple access being developed for probableimplementation is often called Multi-User Superposition Transmission(MUST), but it has various names and different types, including, and notlimited to, Non-Orthogonal Multiple Access (NOMA), Semi-OrthogonalMultiple Access (SOMA), Rate-adaptive constellation Expansion MultipleAccess (EMA), Downlink Multiple User (DL MU), etc. The presentdisclosure is not limited to any of the afore-mentioned technologies,but has wide applicability to any superposition communicationtechnology. Any of these terms as used in this disclosure should beunderstood in their proper context and/or broadest scope.

In general, multiple access superposition refers to communicating tomultiple users by linearly combining amplitude-weighted, encoded, and/ormodulated messages. For example, FIG. 1 has Base Station (BS) 110 andtwo users (or User Equipments (UEs)), a near UE 120 and a far UE 130(“near” and “far” referring to their relative distances from BS 110).Both the near UE 120 and the far UE 130 receive the same signal x,comprising symbol x_(N) for the near UE 120 and symbol x_(F) for far UE130, which can be represented by Equation (1):x=√{square root over (α_(N))}x_(N)+√{square root over (α_(F))}x_(F)  (1)

where α generally refers to transmission power, and thus α_(N) is thetransmission power allocated to the near user signal and α_(F) is thetransmission power allocated to the far user, where α_(N)+α_(F)=1.Sometimes a refers more generally to the ratio of near user power to faruser power, as shown in FIG. 2, which is discussed further below.

Speaking simplistically, near UE 120 decodes symbol x_(F) for far UE 130and uses it to cancel x_(F) as interference, thereby decoding symbolx_(N) intended for the near UE 120. One reiterative process for thistype of cancellation is “Successive Interference Cancellation” or SIC.The far UE 130, on the other hand, simply decodes its own signalx_(F)(although it is possible for the far user to also perform some formof signal cancellation to eliminate x_(N)).

Generally herein, far user symbol x_(F) corresponds to K_(F) bits ofdata represented as (d₀ ^(F)d₁ ^(F) . . . d_(K) _(F) ⁻¹ ^(F)) and nearuser symbol x_(N) corresponds to K_(N) bits of data represented as (d₀^(N)d₁ ^(N) . . . d_(K) _(N) ⁻¹ ^(N)).

FIG. 2 shows an example of a “super-constellation” formed of a (QPSK,QPSK) modulation pair under MUST. “(QPSK, QPSK)” means that both the farand near UE signals are modulated by QPSK. FIG. 2 is the result of adirect symbol mapping (DSM) of QPSK using Equation (1) for both the nearand far users, i.e., a 16-QAM super-constellation. Moreover, in FIG. 2,the constituent x_(F) and x_(N) symbols are separately Gray encoded.

Each of the four bit symbols in the 16-QAM super-constellation in FIG. 2comprises two bits for the symbol intended for the far user and two bitsof the symbol intended for the near user. More specifically, eachfour-bit symbol (b₀, b₁, b₂, b₃) comprises (b₀, b₁)=(d₀ ^(F)d₁ ^(F)),the two bits for the far user, and (b₂, b₃)=(d₀ ^(N)d₁ ^(N)), the twobits for the near user. Thus, the far user constellation is relativelycoarse, because each quadrant represents only one symbol (for example,the upper right quadrant is (00)), while each quadrant of the near userconstellation has all four symbols (00, 01, 10, and 11). However,because the near user is nearer, the near user's received signal isstronger and it will be easier for the near user to distinguish thatlevel of detail than the far user.

In theory, having the near user employ Successive InterferenceCancellation (SIC) by codeword, where the far user codeword is decoded,the original encoded far user codeword reconstructed using the decodedcodeword, and then the reconstructed original signal cancelled from theoverall signal prior to decoding, is optimal in the sense that itachieves capacity.

SUMMARY

Accordingly, the present disclosure has been made to address at leastthe problems and/or disadvantages described above and to provide atleast the advantages described below.

According to one aspect of the present disclosure, a method of powerallocation in a superposition multiple access communication systemcapable of using uniform and non-uniform superposition constellations(super-constellations) is provided, including, for each receiver ireceiving superposition multiple access transmission, calculating theconditional probability P_(c,i) of a bit being correctly received basedon its location within the super-constellation; for each receiver ireceiving superposition multiple access transmission, calculating anormalized weighting coefficient w_(i); calculating the sum S ofweighted spectral efficiencies of all receivers i using the conditionalprobability P_(c,i) and normalized weighting coefficient w_(i) of eachreceiver i; and determining the optimal power allocation α*_(i) forreceiver i by maximizing the sum of weighted spectral efficiencies.

According to another aspect of the present disclosure, a method for auser equipment (UE) is provided, including receiving an indication thatsuperposition transmission is being used to transmit to the UE;receiving an indication of which type of superposition transmission isbeing used to transmit to the UE, wherein at least one type ofsuperposition transmission uses a Gray-mapped Non-uniform-capableConstellation (GNC) super-constellation; and receiving one or moresuperposition transmission parameters, including information concerningpower allocation for the UE, wherein the power allocation for the UE wasdetermined by: calculating the conditional probability of a bit beingcorrectly received by the UE based on its location within thesuper-constellation; calculating a normalized weighting coefficient forthe UE; calculating the sum of weighted spectral efficiencies of all UEsreceiving superposition transmission using the conditional probabilitiesand normalized weighting coefficients of the all UEs; and determiningthe optimal power allocation for the UE by maximizing the sum ofweighted spectral efficiencies.

According to yet another aspect of the present disclosure, an apparatusis provided for power allocation in a superposition multiple accesscommunication system capable of using uniform and non-uniformsuperposition constellations (super-constellations), including at leastone non-transitory computer-readable medium storing instructions capableof execution by a processor; and at least one processor capable ofexecuting instructions stored on the at least one non-transitorycomputer-readable medium, where the execution of the instructionsresults in the apparatus performing a method including, for eachreceiver i receiving superposition multiple access transmission,calculating the conditional probability P_(c,i) of a bit being correctlyreceived based on its location within the super-constellation; for eachreceiver i receiving superposition multiple access transmission,calculating a normalized weighting coefficient w_(i); calculating thesum S of weighted spectral efficiencies of all receivers i using theconditional probability P_(c,i) and normalized weighting coefficientw_(i) of each receiver i; and determining the optimal power allocationα*_(i) for receiver i by maximizing the sum of weighted spectralefficiencies.

According to still yet another aspect of the present disclosure, a userequipment (UE) is provided, including at least one non-transitorycomputer-readable medium storing instructions capable of execution by aprocessor; and at least one processor capable of executing instructionsstored on the at least one non-transitory computer-readable medium,where the execution of the instructions results in the UE performing amethod including receiving an indication that superposition transmissionis being used to transmit to the UE; receiving an indication of whichtype of superposition transmission is being used to transmit to the UE,wherein at least one type of superposition transmission uses aGray-mapped Non-uniform-capable Constellation (GNC) super-constellation;and receiving one or more superposition transmission parameters,including information concerning power allocation for the UE, whereinthe power allocation for the UE was determined by: calculating theconditional probability of a bit being correctly received by the UEbased on its location within the super-constellation; calculating anormalized weighting coefficient for the UE; calculating the sum ofweighted spectral efficiencies of all UEs receiving superpositiontransmission using the conditional probabilities and normalizedweighting coefficients of the all UEs; and determining the optimal powerallocation for the UE by maximizing the sum of weighted spectralefficiencies.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other aspects, features, and advantages of certainembodiments of the present disclosure will be more apparent from thefollowing detailed description, taken in conjunction with theaccompanying drawings, in which:

FIG. 1 is a diagram of an example of Multi-User SuperpositionTransmission (MUST), with both a near UE and a far UE sharing asuperposed signal;

FIG. 2 is a mapping of a super-constellation formed by direct symbolmapping (DSM) of a (QPSK, QPSK) modulation pair for a far user and anear user;

FIG. 3A is the mapping of a special case of Gray-mappedNon-uniform-capable Constellation (GNC) for a (QPSK, QPSK) modulationpair where the lattice is uniform, according to an embodiment of thepresent disclosure;

FIG. 3B is the mapping of a non-uniform GNC for a (QPSK, QPSK)modulation pair according to various embodiments of the presentdisclosure;

FIG. 4 is a mapping of a (16-QAM, QPSK) GNC super-constellationaccording to an embodiment of the present disclosure;

FIG. 5 is a mapping of a (QPSK, 16-QAM) GNC super-constellationaccording to an embodiment of the present disclosure;

FIG. 6 is a flowchart of a general method of power allocation accordingto an embodiment of the present disclosure, and

FIG. 7 is a flowchart of a more specific method of power allocationaccording to an embodiment of the present disclosure.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE PRESENT DISCLOSURE

Hereinafter, embodiments of the present disclosure are described indetail with reference to the accompanying drawings. It should be notedthat the same elements will be designated by the same reference numeralsalthough they are shown in different drawings. In the followingdescription, specific details such as detailed configurations andcomponents are merely provided to assist the overall understanding ofthe embodiments of the present disclosure. Therefore, it should beapparent to those skilled in the art that various changes andmodifications of the embodiments described herein may be made withoutdeparting from the scope and spirit of the present disclosure. Inaddition, descriptions of well-known functions and constructions areomitted for clarity and conciseness. The terms described below are termsdefined in consideration of the functions in the present disclosure, andmay be different according to users, intentions of the users, orcustoms. Therefore, the definitions of the terms should be determinedbased on the contents throughout the specification.

The present disclosure may have various modifications and variousembodiments, among which embodiments are described below in detail withreference to the accompanying drawings. However, it should be understoodthat the present disclosure is not limited to the embodiments, butincludes all modifications, equivalents, and alternatives within thespirit and the scope of the present disclosure.

Although the terms including an ordinal number such as first, second,etc. may be used for describing various elements, the structuralelements are not restricted by the terms. The terms are only used todistinguish one element from another element. For example, withoutdeparting from the scope of the present disclosure, a first structuralelement may be referred to as a second structural element. Similarly,the second structural element may also be referred to as the firststructural element. As used herein, the term “and/or” includes any andall combinations of one or more associated items.

The terms used herein are merely used to describe various embodiments ofthe present disclosure but are not intended to limit the presentdisclosure. Singular forms are intended to include plural forms unlessthe context clearly indicates otherwise. In the present disclosure, itshould be understood that the terms “include” or “have” indicateexistence of a feature, a number, a step, an operation, a structuralelement, parts, or a combination thereof, and do not exclude theexistence or probability of addition of one or more other features,numerals, steps, operations, structural elements, parts, or combinationsthereof.

Unless defined differently, all terms used herein have the same meaningsas those understood by a person skilled in the art to which the presentdisclosure belongs. Such terms as those defined in a generally useddictionary are to be interpreted to have the same meanings as thecontextual meanings in the relevant field of art, and are not to beinterpreted to have ideal or excessively formal meanings unless clearlydefined in the present disclosure.

Broadly speaking, the present disclosure provides guidance, within asuperposition multiple access communication context, on (1) how to splittransmit power between near and far users using weighted spectralefficiencies; and (2) how to determine more detailed and more efficientbit-swapping rules.

Gray-Mapped Non-Uniform-Capable Constellation (GNC)

A related non-provisional patent application by the same inventors,entitled Apparatus and Method for Superposition Transmission, was filedon Jan. 15, 2016 and given U.S. application Ser. No. 14/997,106, andclaims priority to the same four U.S. provisional patent applications asdoes the present application. That application is expressly incorporatedby reference in its entirety.

In U.S. application Ser. No. 14/997,106 and similarly-incorporated U.S.Prov. Pat. App. Ser. No. 62/173,241 and 62/203,818 to which it claimspriority (referred to jointly herein as “the other application”), a newtype of superposition super-constellation is described: the Gray-mappedNon-uniform Constellation or Gray-mapped Non-uniform-capableConstellation (GNC). Besides being gray encoded, a GNCsuper-constellation can have unequal spaces between neighboring symbols(a feature which will be exploited herein), is formed by a direct-sum ofregularly spaced lattices, which leads to simplified jointlog-likelihood ratio (LLR) generation, and can be easily extended tomore than two users (i.e., more than simply a “near” and a “far” user).

Among other things, the other application also discussed that Graymapping may not be guaranteed under certain power-splitting conditions,although using a GNC super-constellation under some of thosecircumstances would still provide advantages over doing otherwise.Moreover, a “bit swapping” technique was discussed, where, under certainconditions, the bits of the near and far users are swapped within theGNC to provide better results. The specific results are summarized inTable 1 below:

TABLE 1 α_(F) Regions for GNC With and Without Bit-Swapping “Far” UE“Near” UE Resulting “Super- constellation constellation constellation”Bit- (2^(K) ^(F) )-QAM (2^(K) ^(N) )-QAM (2^(K) ^(F) ^(+K) ^(N) )-QAMSwapped Single Layer Two Layers QPSK QPSK 16-QAM OFF α_(F) ≥ 0.5 α_(F) ≥0.3333 ON α_(F) ≤ 0.5 α_(F) ≤ 0.3333 16-QAM QPSK 64-QAM OFF α_(F) ≥0.6429 α_(F) ≥ 0.4737 ON α_(F) ≤ 0.1667 α_(F) ≤ 0.0909 QPSK 16-QAM64-QAM OFF α_(F) ≥ 0.8333 α_(F) ≥ 0.7143 ON α_(F) ≤ 0.3571 α_(F) ≤0.2174 16-QAM 16-QAM 256-QAM OFF α_(F) ≥ 0.9 α_(F) ≥ 0.8182 ON α_(F) ≤0.1 α_(F) ≤ 0.0526 64-QAM QPSK 256-QAM OFF α_(F) ≥ 0.7 α_(F) ≥ 0.5385 ONα_(F) ≤ 0.0455 α_(F) ≤ 0.0233 QPSK 64-QAM 256-QAM OFF α_(F) ≥ 0.9545α_(F) ≥ 0.9130 ON α_(F) ≤ 0.3 α_(F) ≤ 0.1795

Thus, when forming a (16-QAM, QPSK) super-constellation in a singlelayer environment, if α_(F)≥0.6429, a GNC with no bit-swapping should beused. If α_(F)≤0.1667, a GNC with bit-swapping should be used. Thismeans there is an “exclusion zone” between 0.1667 and 0.6429 where noGNC/bit-swapping combination works well, and other methods forgenerating super-constellations might be used there.

Although the other application provides guidance for defining exclusionzones and regions for performing bit-swapping or not, no furtherguidance is provided for optimization within those zones. In otherwords, for example, a single layer (16-QAM, QPSK) GNC with nobit-swapping should be used when α_(F)≥0.6429, but the other applicationprovides no guidance on the optimal value for α_(F) within that region.

This application provides guidance concerning the optimal powerdistribution and the optimal bit-swapping rules.

I. Optimal Power Distribution Using GNC with or without Bit-Swapping

Power distribution herein is optimized for GNC super-constellations bythe use of weighted spectral efficiencies which take into account themodulation and coding schemes (MCS), symbol error rates, decoding errorrates, and/or bias terms adjusting for coding gains and bit locations.In this embodiment, the sum of weighted spectral efficiencies is used,but, in other embodiments, other appropriate spectral efficiency metricscould be used, such as the weighted average of spectral efficiencies,for example.

Taking the simplest example, where there is only a near and a far user,only the inner bits of the super-constellation are of concern for thenear user, while only the outer bits of the super-constellation are ofconcern to the far user. Thus, a single super-constellation provides twoprobabilities for a near or far user symbol being correct. For thissituation, the sum of weighted spectral efficiencies is expressed for anear and far user as Equation (2)(a):S=w _(F) P _(c,F) +w _(N) P _(c,N)  (2)(a)

or more generally as

$S = {\sum\limits_{i = 1}^{K}\;{w_{i}P_{c,i}}}$for K users, where the probability P_(c,i) of a detected symbol beingcorrect is defined as Equation (2)(b):P _(c,i)=Σ_(k=1) ^(M) P({circumflex over (x)} _(k,i) =x _(k,i))  (2)(b)

where {circumflex over (x)}_(k,i) denotes the detected symbol at the kthsymbol for user/UE i. Thus, P_(c,i) handles uncoded rates. The code rateand its corresponding coding gains are captured with weightingcoefficients. The maximum effective amount of bits to be carried on foruser/UE i is expressed as C_(i) log₂ M_(i). Thus, the weightingcoefficient for user/UE i, W_(i), can be normalized with bias terms, asshown by Equation (2)(c):

$\begin{matrix}{w_{i} = \frac{{c_{i}\log_{2}M_{i}} + {\Delta_{i}\left( {c_{i},s_{i}} \right)}}{\sum_{k}\left( {{c_{k}\log_{2}M_{k}} + {\Delta_{k}\left( {c_{k},s_{k}} \right)}} \right)}} & {(2)(c)}\end{matrix}$

where:

C_(i) is the code rate for user/UEi;

S_(i) is a flag indicating whether UE i's bits are swapped or not; and

Δ_(i) (C_(i), S_(i)) is a bias term to compensate for the effect ofcoding gains between inner and outer bits, and is a function of C_(i)and S_(i).

The bias term Δ_(i)(C_(i), S_(i)) is needed because, although C_(i) log₂M_(i) represents the maximum amount of bits that can be transmitted persymbol, the effective amount of bits to be carried are changed due to:

-   -   the effect of coding gains is not linear, and    -   the bit location changes the effective coding gain.

In general, the outer bits are more robust than the inner bits such thatthe effective coding gain, even if at the same coding rate, can bedifferent depending on bit position, which affects the block error rate(BLER). Accordingly, the bias term Δ_(i) (C_(i), S_(i)) is added tocompensate for both impacts on the bit domain. For example, if the sameMCS is used for both the far and near user, additional bits would beadded to the near user's Δ_(i)(C_(i), S_(i)) to balance the performance.As another example, a high code rate having less decoding gains wouldneed a high Δ_(i) (C_(i), S_(i)) to compensate for the effect on the bitpositions.

The bias terms Δ_(i) (C_(i), S_(i)) could be prepared offline and savedas a lookup table (LUT). The optimal α*_(F) to maximize the sum ofweighted spectral efficiencies for both users/UEs can be calculatedgenerally by Equation (3), as will be shown in detail below:α*_(F)=arg max_(α) _(F) _(≥α) _(F,th) S  (3)

where α_(F,th) follows Table 1 above, and is determined depending on themodulation combination. For example, when a single stream is used withno-swapping, then α_(F,th) should be 0.5 with (QPSK, QPSK) as shown inthe first column in Table 1.

Since C is changed with the signal-to-noise ratio (SNR), the powerdistribution coefficient α*_(F) varies in a corresponding manner and, asmentioned above, it would vary depending on whether bit-swapping wasused. Lastly, P_(c,i) depends on using the GNC system as described hereand in the other application.

As discussed further below, the idea of using weighted spectralefficiencies could be extended to a scheme using codeword leveldecoding.

A. Optimal α_(F) for (QPSK, QPSK) GNC Super-Constellation

(1) Uniform (QPSK, QPSK) GNC (FIG. 3A)

To explain the idea, consider FIG. 3A, which is a single-layer (QPSK,QPSK) GNC super-constellation where α_(N)=0.20 and α_(F)=0.80 or,equivalently, p=q=1. The parameters q and p mentioned above are newvariables used for generating the GNC super-constellation, as discussedin more detail in the other application. Generally speaking, qguarantees the desired power split between the users and p relates tounit constellation power.

FIG. 3A is a special case, where the GNC super-constellation forms auniform 16-QAM lattice (instead of a non-uniform lattice, as discussedbelow in reference to FIG. 3B). Uniform or not, the four feasible realvalues for the points are −p(2+q), −p(2−q), p(2−q), and p(2+q), as shownin FIGS. 3A and 3B, and the values on the imaginary axis are symmetricto these real values.

In FIG. 3A, bits (b₀, b₁) divide the constellation into four groups,where each group belongs in one quadrant in the (x₁, x_(Q)) coordinatesystem. Bits (b₂, b₃) define Gray labeled constellation points with eachset for a given value of the pair (b₀, b₁). In other words, the pairs ofbits (b₀, b₁) and (b₂, b₃) form a nested structure where (b₀, b₁)constitute the “outer” part of the direct sum and (b₂, b₃) form the“inner” part of the direct sum, as shown in Equation (4).

$\begin{matrix}{\left( {b_{0},b_{1},b_{2},b_{3}} \right) = {\underset{({{outer}\mspace{14mu}{part}})}{\left( {b_{0},b_{1}} \right)} \oplus \underset{({{inner}\mspace{14mu}{part}})}{\left( {b_{2},b_{3}} \right)}}} & (4)\end{matrix}$

The outer bits are assigned to the far user, (b₀, b₁)=(d₀ ^(F)d₁ ^(F)),and the inner bits are assigned to the near user, (b₂, b₃)=(d₀ ^(N)d₁^(N)). Assuming the received power is P, and the power constraint valueC is 10/P, the unequal power split can be made part of symbol mapping asshown in Equations (5)(a) and (5)(b) below:

$\begin{matrix}{x = {\frac{1}{\sqrt{C}}\left\{ {{{p\left( {1 - {2\; b_{0}}} \right)}\left\lbrack {2 - {q\left( {1 - {2\; b_{2}}} \right)}} \right\rbrack} + {j\;{{p\left( {1 - {2\; b_{1}}} \right)}\left\lbrack {2 - {q\left( {1 - {2\; b_{3}}} \right)}} \right\rbrack}}} \right\}}} & {(5)(a)}\end{matrix}$

which is the same as

$\begin{matrix}{x = {\frac{1}{\sqrt{C}}\left\{ {{{p\left( {1 - {2\; d_{0}^{F}}} \right)}\left\lbrack {2 - {q\left( {1 - {2\; d_{0}^{N}}} \right)}} \right\rbrack} + {j\;{{p\left( {1 - {2\; d_{1}^{F}}} \right)}\left\lbrack {2 - {q\left( {1 - {2\; d_{1}^{N}}} \right)}} \right\rbrack}}} \right\}}} & {(5)(b)}\end{matrix}$

where p and q are positive real-valued numbers, as mentioned above,which, when in a one-layer or scalar environment (or “Scenario 1” asestablished by 3GPP RAN1 to evaluate MUST), are subject to the followingconstraints:

$\begin{matrix}{{2{p^{2}\left( {4 + q^{2}} \right)}} = C} & {(6)(a)} \\{\frac{q^{2}}{4} = \frac{1 - \alpha_{F}}{\alpha_{F}}} & {(6)(b)}\end{matrix}$

where Equation (6)(a) arises from the total power requirement or unitconstellation power and Equation (6)(b) arises from the power splitrequirement between the near and far users, or, equivalently, the splitbetween the (b₀, b₁) bits and (b₂, b₃) bits.

As discussed above, FIG. 3A is the special case of (QPSK, QPSK) GNCsuper-constellation where α_(N)=0.20 and α_(F)=0.80 or, equivalently,p=q=1, thereby forming a uniform 16-QAM lattice.

(2) Non-uniform (QPSK,QPSK) GNC (FIG. 3B)

FIG. 3B is a non-uniform (QPSK, QPSK) GNC super-constellation havingunequal distances between constellation points. Specifically, thenon-uniform GNC super-constellation as shown in FIG. 3B has twodifferent values for the distance between symbols, which are:d _(min,1)=2p(2−q)  (7)(a)d_(min,2)=2pq  (7)(b)

For simplicity, the nearest neighbor symbols can only be considered topick erroneous symbols up. As the distance increases, the errorprobability exponentially decreases. In addition, as SNR increases, thevalue of Q function significantly decreases. Thus, other symbols exceptfor the nearest neighbor symbols would have small impact on the proposedpower allocation.

In general, constellation points may be grouped in a number of ways interms of defining the conditional probability, such as shown in theparticular example of Equation (8)(a) below, as would be understood byone of ordinary skill in the art. Points within the same distance andhaving the same number of different bits could be grouped, for example.

For a far user bit (which is also an outer bit, like all far user bits),the conditional probability of being correct is given by Equation(8)(a):

$\begin{matrix}{{P\left( {{\hat{x}}_{k,F} = {\left. x_{k,F} \middle| x_{F} \right. = x_{k,F}}} \right)} = \left\{ \begin{matrix}1 & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 4\mspace{14mu}{corner}\mspace{14mu}{points}} \\\left( {1 - Q_{1}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 4\mspace{14mu}{inner}\mspace{14mu}{points}} \\\left( {1 - Q_{1}} \right) & {{if}\mspace{20mu} x_{k}\mspace{11mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 8\mspace{14mu}{edge}\mspace{14mu}{points}}\end{matrix} \right.} & {(8)(a)}\end{matrix}$

where Q_(i) is defined as a Q-function

${Q(x)} = {\frac{1}{\sqrt{2\pi}}{\int_{x}^{\infty}{{\exp\left( {- \frac{u^{2}}{2}} \right)}\ {\mathbb{d}u}}}}$with the argument d_(min,i)/2σ. The Q-function indicates the tailprobability of the standard normal distribution. Thus, Equation (8)(a)can be re-written as Equation (8)(b):

$\begin{matrix}{P_{c,F} = {{P\left( {{\hat{x}}_{F} = x_{F}} \right)} = {{\frac{8}{16}\left( {1 - Q_{1}} \right)} + {\frac{4}{16}\left( {1 - Q_{1}} \right)^{2}} + \frac{4}{16}}}} & {(8)(b)}\end{matrix}$

For a near user bit (which is also an inner bit, like all near userbits), the conditional probability of being correct is given by Equation(9)(a):P({circumflex over (x)} _(k,N) =x _(k,N) |x _(N) =x _(k,N))=(1−Q₂)²  (9)(a)

for all points in the super-constellation. Since each constellationpoint is also uniformly selected, the conditional probability can alsobe given by Equation (9)(b):P _(c,N) =P({circumflex over (x)} _(N) =x _(N))=(1−Q ₂)²  (9)(b)

Therefore, the sum of weighted spectral efficiencies for the (QPSK,QPSK) GNC super-constellation is given by applying Equation (2)(a) tothe case of a near and far user, as shown in Equation (10):

$\begin{matrix}\begin{matrix}{S = {{w_{F}P_{c,F}} + {w_{N}P_{c,N}}}} \\{= {{w_{F}\left( {{\frac{1}{2}\left( {1 - Q_{1}} \right)} + {\frac{1}{4}\left( {1 - Q_{1}} \right)^{2}} + \frac{1}{4}} \right)} + {w_{N}\left( {1 - Q_{2}} \right)}^{2}}}\end{matrix} & (10)\end{matrix}$

where (w_(F), w_(N)) is a set of normalizing weighting coefficients asin Equation (2)(c). The optimal α*_(F) to maximize the sum of weightedspectral efficiencies for both users/UEs can be selected by Equation(11)(a):α*_(F)=arg max_(α) _(F) _(≥0.5)S  (11)(a)

Since the combination of (QPSK, QPSK) is symmetric over bit-swapping,the optimal α*_(F) for the bit-swapped GNC, denoted as α⁻*_(F), is givenby Equation (11)(b):α⁻*_(F)=1−α*_(F)  (11)(b)

B. Optimal (α_(F) for (16-QAM, QPSK) GNC Super-Constellation (FIG. 4)

FIG. 4 shows a single-layer (16-QAM, QPSK) GNC super-constellation withno bit-swapping, where α_(N)=0.20 and α_(F)=0.80, for which the optimalpower distribution is derived below according to an embodiment of thepresent disclosure in terms of the sums of spectral efficiencies. Thereal parts of the symbol mapping are identified in terms of p and q atthe bottom of FIG. 4.

The bit assignment is (b₀, b₁, b₂, b₃, b₄, b₅)=(d₀ ^(F), d₁ ^(F), d₀^(N), d₁ ^(N), d₂ ^(N), d₃ ^(N)) (except when the bits are swapped).When the received power is P, the power constraint value C is 42/P, andthe symbol mapping process is as shown in Equation (12) below:

$\begin{matrix}{x = {{\frac{1}{\sqrt{C}}\left( {1 - {2b_{0}}} \right){p\left( {4 - {{q\left( {1 - {2b_{2}}} \right)}\left( {2 - \left( {1 - {2b_{4}}} \right)} \right)}} \right)}} + {j\frac{1}{\sqrt{C}}\left( {1 - {2b_{1}}} \right){p\left( {4 - {{q\left( {1 - {2b_{3}}} \right)}\left( {2 - \left( {1 - {2b_{5}}} \right)} \right)}} \right)}}}} & (12)\end{matrix}$

which generates four positive real values: p(4−3q), p(4−q), p(4+q), andp(4+3q). In a one-layer or scalar environment (or “Scenario 1” asestablished by 3GPP RAN1 to evaluate MUST), p, q are subject to thefollowing constraints:

$\begin{matrix}{{2{p^{2}\left( {16 + {5q^{2}}} \right)}} = C} & {(13)(a)} \\{\frac{10q^{2}}{32} = \frac{1 - \alpha_{F}}{\alpha_{F}}} & {(13)(b)}\end{matrix}$

where again the former comes from the total power requirement/unitconstellation power and the latter arises from the power splitrequirement between the near and far users. Note that setting p=q=1 orequivalently α_(F)=20/21 results in the uniform 64-QAM constellation.

Contrary to the uniform constellation, the non-uniform GNCsuper-constellation as shown in FIG. 4 has two different values for thedistance between symbols, which are:d _(min,1)=2p(4−3q)  (14)(a)d_(min,2)=2pq  (14)(b)

For a far user, the conditional probability of being correct can belisted as Equation (15)(a):

$\begin{matrix}{{P\left( {{\hat{x}}_{k,F} = {\left. x_{k,F} \middle| x_{F} \right. = x_{k,F}}} \right)} = \left\{ \begin{matrix}\left( {1 - Q_{1}} \right) & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 24\mspace{14mu}{edge}\text{/}{inner}\mspace{14mu}{points}} \\\left( {1 - Q_{1}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 4\mspace{14mu}{inner}\mspace{14mu}{points}} \\1 & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 36\mspace{14mu}{outer}\mspace{14mu}{points}}\end{matrix} \right.} & {(15)(a)}\end{matrix}$

which only takes the nearest neighbor symbols into account. Each symbolis equally selected with a probability 1/64such that the probability ofbeing correct can be calculated as:

$\begin{matrix}{P_{c,F} = {{P\left( {{\hat{x}}_{F} = x_{F}} \right)} = {{\frac{3}{8}\left( {1 - Q_{1}} \right)} + {\frac{1}{16}\left( {1 - Q_{1}} \right)^{2}} + \frac{9}{16}}}} & {(15)(b)}\end{matrix}$

The conditional probability for the far user can be re-listed, using theadditional distances of:d _(min,3)=2d _(min,1) +d _(min,2)  (15)(c)d _(min,4) =d _(min,1)+2d _(min,2)  (15)(d)

to arrive at Equation (15)(e):

$\begin{matrix}{{P\left( {{\hat{x}}_{k,F} = {\left. x_{k,F} \middle| x_{F} \right. = x_{k,F}}} \right)} = \left\{ \begin{matrix}\left( {1 - Q_{1}} \right) & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 16\mspace{14mu}{edge}\text{/}{inner}\mspace{14mu}{points}} \\{\left( {1 - Q_{1}} \right)\left( {1 - Q_{4}} \right)} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 8\mspace{14mu}{out}\text{/}{inner}\mspace{14mu}{points}} \\\left( {1 - Q_{1}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 4\mspace{14mu}{inner}\mspace{14mu}{points}} \\\left( {1 - Q_{4}} \right) & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 16\mspace{14mu}{out}\text{/}{inner}\mspace{14mu}{points}} \\\left( {1 - Q_{4}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 4\mspace{14mu}{out}\text{/}{inner}\mspace{14mu}{points}} \\1 & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 16\mspace{14mu}{outer}\mspace{14mu}{points}}\end{matrix} \right.} & {(15)(e)}\end{matrix}$

which updates Equation (15)(b) to Equation (15)(f):

$\begin{matrix}\begin{matrix}{P_{c,F} = {P\left( {{\hat{x}}_{F} = x_{F}} \right)}} \\{= {{\frac{1}{4}\left( {1 - Q_{1}} \right)} + {\frac{1}{8}\left( {1 - Q_{1}} \right)\left( {1 - Q_{4}} \right)} + {\frac{1}{16}\left( {1 - Q_{1}} \right)^{2}} +}} \\{{\frac{1}{4}\left( {1 - Q_{4}} \right)} + {\frac{1}{16}\left( {1 - Q_{4}} \right)^{2}} + \frac{1}{4}}\end{matrix} & {(15)(f)}\end{matrix}$

For the near user, the conditional probability of being correct can belisted as Equation (16)(a):

$\begin{matrix}{{P\left( {{\hat{x}}_{k,N} = {\left. x_{k,N} \middle| x_{N} \right. = x_{k,N}}} \right)} = \left\{ \begin{matrix}\left( {1 - Q_{2}} \right)^{2} & \begin{matrix}{{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 4\mspace{14mu}{c{orner}}\text{/}4\mspace{14mu}{inner}\text{/}} \\{8\mspace{14mu}{edge}\mspace{14mu}{points}}\end{matrix} \\{\left( {1 - {2Q_{2}}} \right)\left( {1 - Q_{2}} \right)} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 32\mspace{14mu}{e{dge}}\text{/}{inner}\mspace{14mu}{points}} \\\left( {1 - {2Q_{2}}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 16\mspace{14mu}{out}\text{-}{inner}\mspace{14mu}{points}}\end{matrix} \right.} & {(16)(a)}\end{matrix}$

for all points on a uniform constellation map. Similarly, theprobability of being correct can be calculated as:

$\begin{matrix}\begin{matrix}{P_{c,N} = {P\left( {{\hat{x}}_{N} = x_{N}} \right)}} \\{= {{\frac{1}{4}\left( {1 - Q_{2}} \right)^{2}} + {\frac{1}{2}\left( {1 - {2Q_{2}}} \right)\left( {1 - Q_{2}} \right)} + {\frac{1}{4}\left( {1 - {2Q_{2}}} \right)^{2}}}}\end{matrix} & {(16)(b)}\end{matrix}$

The conditional probability for the near user can be re-listed, usingthe additional distance of d_(min,3) to Equation (16)(c):

$\begin{matrix}{{P\left( {{\hat{x}}_{k,N} = {\left. x_{k,N} \middle| x_{N} \right. = x_{k,N}}} \right)} = \left\{ \begin{matrix}\left( {1 - Q_{2}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 4\mspace{14mu}{corner}\mspace{14mu}{points}} \\{\left( {1 - Q_{2}} \right)\left( {1 - {2Q_{2}}} \right)} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 16\mspace{14mu}{e{dge}}\text{/}{inner}\mspace{14mu}{points}} \\\left( {1 - {2Q_{2}}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 16\mspace{14mu}{out}\text{/}{inner}\mspace{14mu}{points}} \\{\left( {1 - Q_{2}} \right)\left( {1 - Q_{2} - Q_{3}} \right)} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 8\mspace{14mu}{e{dge}}\text{/}{inner}\mspace{14mu}{points}} \\{\left( {1 - {2Q_{2}}} \right)\left( {1 - Q_{2} - Q_{3}} \right)} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 16\mspace{14mu}{out}\text{/}{inner}\mspace{14mu}{points}} \\\left( {1 - Q_{2} - Q_{3}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 4\mspace{14mu}{inner}\mspace{14mu}{points}}\end{matrix} \right.} & {(16)(c)}\end{matrix}$

which updates Equation (16)(b) to Equation (16)(d):

$\begin{matrix}\begin{matrix}{P_{c,N} = {P\left( {{\hat{x}}_{N} = x_{N}} \right)}} \\{= {{\frac{1}{16}\left( {1 - Q_{2}} \right)^{2}} + {\frac{1}{4}\left( {1 - Q_{2}} \right)\left( {1 - {2Q_{2}}} \right)} + {\frac{1}{4}\left( {1 - {2Q_{2}}} \right)^{2}} +}} \\{{\frac{1}{8}\left( {1 - Q_{2}} \right)\left( {1 - Q_{2} - Q_{3}} \right)} + {\frac{1}{4}\left( {1 - {2Q_{2}}} \right)\left( {1 - Q_{2} - Q_{3}} \right)} +} \\{\frac{1}{16}\left( {1 - Q_{2} - Q_{3}} \right)^{2}}\end{matrix} & {(16)(d)}\end{matrix}$

Therefore, the sum of weighted spectral efficiencies for the (16-QAM,QPSK) GNC super-constellation is Equation (17):

$\begin{matrix}\begin{matrix}{S = {{w_{F}P_{c,F}} + {w_{N}P_{c,N}}}} \\{= {{w_{F}\left( {{\frac{3}{4}\left( {1 - Q_{1}} \right)} + {\frac{1}{8}\left( {1 - Q_{1}} \right)^{2}}} \right)} +}} \\{w_{N}\left( {{\frac{1}{4}\left( {1 - Q_{2}} \right)^{2}} + {\frac{1}{2}\left( {1 - {2Q_{2}}} \right)\left( {1 - Q_{2}} \right)} + {\frac{1}{4}\left( {1 - {2Q_{2}}} \right)^{2}}} \right)}\end{matrix} & (17)\end{matrix}$

where (w_(F), w_(N)) is a set of normalizing weighting coefficient as inEquation (5)(c). Alternatively, it could be updated by using Equation(15)(e) or (16)(d) Thus, the optimal α*_(F) for the (16-QAM, QPSK) GNCsuper-constellation can be selected as Equation (18):α*_(F)=arg max_(α) _(F) _(≥0.65)S  (18)

C. Optimal α_(F) for (QPSK, 16-QAM) GNC Super-Constellation (FIG. 5)

FIG. 5 shows a (QPSK, 16-QAM) GNC super-constellation with nobit-swapping, where α_(F)=0.90. Its optimal power distribution in termsof the sums of spectral efficiencies is derived below according to anembodiment of the present disclosure.

The outer bits and the inner bits are positioned for the far user andthe near user, respectively. Thus, 6 bits per symbol are concatenated to(b₀, b₁, b₂, b₃, b₄, b₅)=(d₀ ^(F), d₁ ^(F), d₂ ^(F), d₃ ^(F), d₀ ^(N),d₁ ^(N)), and each symbol is mapped with p and q to Equation (19):

$\begin{matrix}{x = {{\frac{1}{\sqrt{C}}\left( {1 - {2b_{0}}} \right){p\left( {4 - {\left( {1 - {2b_{2}}} \right)\left( {2 - {q\left( {1 - {2b_{4}}} \right)}} \right)}} \right)}} + {j\frac{1}{\sqrt{C}}\left( {1 - {2b_{1}}} \right){p\left( {4 - {\left( {1 - {2b_{3}}} \right)\left( {2 - {q\left( {1 - {2b_{5}}} \right)}} \right)}} \right)}}}} & (19)\end{matrix}$

where C=42 for the unit power constraint. When the received power is P,the power constraint value C is 42/P. For N_(s)=1, p, q are subject tothe following constraints:

$\begin{matrix}{{2{p^{2}\left( {20 + q^{2}} \right)}} = C} & {(20)(a)} \\{\frac{q^{2}}{20} = \frac{1 - \alpha_{F}}{\alpha_{F}}} & {(20)(b)}\end{matrix}$

As mentioned above, the power distribution between α_(N) and α_(F) isdetermined by q. Note that setting p=q=1 or equivalently α_(F)=16/21results in the uniform 64-QAM constellation. As shown by FIG. 5, the twodifferent distances between symbols are:d _(min,1)=2p(2−q)  (21)(a)d_(min,2)=2pq  (21)(b)

which are used to derive an error rate for both the far and nearusers/UEs, respectively.

For a far user, the conditional probability of being correct with thenearest neighbor symbols can be listed as Equation (22)(a):

$\begin{matrix}{{P\left( {{\hat{x}}_{k,F} = {{x_{k,F}❘x_{F}} = x_{x,F}}} \right)} = \left\{ \begin{matrix}\left( {1 - Q_{1}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 36\mspace{14mu}{inner}\mspace{14mu}{points}} \\\left( {1 - Q_{1}} \right) & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 24\mspace{14mu}{edge}\mspace{14mu}{points}} \\1 & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 4\mspace{14mu}{corner}\mspace{14mu}{points}}\end{matrix} \right.} & {(22)(a)}\end{matrix}$

such that the probability of being correct is given by:

$\begin{matrix}{P_{c,F} = {{P\left( {{\hat{x}}_{F} = x_{F}} \right)} = {{\frac{3}{8}\left( {1 - Q_{1}} \right)} + {\frac{9}{16}\left( {1 - Q_{1}} \right)^{2}} + \frac{1}{16}}}} & {(22)(b)}\end{matrix}$

and, given the 3^(rd) and 4^(th) distances of:d _(min,3)=2d _(min,1) +d _(min,2)  (22)(c)d _(min,4) =d _(min,1)+2d _(min,2)  (22)(d)

the conditional probability for the far user can be elaborated as:

$\begin{matrix}{{P\left( {{\hat{x}}_{k,F} = {{x_{k,F}❘x_{F}} = x_{x,F}}} \right)} = \left\{ \begin{matrix}\left( {1 - Q_{4}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 4\mspace{14mu}{corner}\mspace{14mu}{points}} \\{\left( {1 - Q_{1}} \right)\left( {1 - Q_{4}} \right)} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 8\mspace{14mu}{edge}\mspace{14mu}{points}} \\{\left( {1 - Q_{1} - Q_{4}} \right)\left( {1 - Q_{4}} \right)} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 16\mspace{14mu}{edge}\mspace{14mu}{points}} \\\left( {1 - Q_{1}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 4\mspace{14mu}{inner}\mspace{14mu}{points}} \\{\left( {1 - Q_{1}} \right)\left( {1 - Q_{1} - Q_{4}} \right)} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 16\mspace{14mu}{inner}\mspace{14mu}{points}} \\\left( {1 - Q_{1} - Q_{4}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 16\mspace{14mu}{inner}\mspace{14mu}{points}}\end{matrix} \right.} & {(22)(e)}\end{matrix}$

which updates the conditional probability of Equation (22)(b) toEquation (22)(f):

$\begin{matrix}\begin{matrix}{P_{c,F} = {P\left( {{\hat{x}}_{F} = x_{F}} \right)}} \\{= {{\frac{1}{16}\left( {1 - Q_{4}} \right)^{2}} + {\frac{1}{8}\left( {1 - Q_{1}} \right)\left( {1 - Q_{4}} \right)} +}} \\{{\frac{1}{4}\left( {1 - Q_{1} - Q_{4}} \right)\left( {1 - Q_{4}} \right)} +} \\{{\frac{1}{16}\left( {1 - Q_{1}} \right)^{2}} + {\frac{1}{4}\left( {1 - Q_{1}} \right)\left( {1 - Q_{1} - Q_{4}} \right)} + {\frac{1}{4}\left( {1 - Q_{1} - Q_{4}} \right)^{2}}}\end{matrix} & {(22)(f)}\end{matrix}$

For a near user, the conditional probability of being correct is givenby Equation (23)(a):P({circumflex over (x)} _(k,N) =x _(k,N) |x _(N) =x _(k,N))=(1−Q₂)²  (23)(a)

for all points on a uniform constellation map. All points are assignedwith an equal probability so that:P _(c,N) =P({circumflex over (x)} _(N) =x _(N))=(1−Q ₂)²  (23)(b)

as with the far user, the conditional probability for the near user canbe elaborated with Q₃ as:

$\begin{matrix}{{P\left( {{\hat{x}}_{k,N} = {{x_{k,N}❘x_{N}} = x_{x,N}}} \right)} = \left\{ \begin{matrix}\left( {1 - Q_{2}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 4\mspace{14mu}{corner}\mspace{14mu}{points}} \\{\left( {1 - Q_{2}} \right)\left( {1 - Q_{2} - Q_{3}} \right)} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 24\mspace{14mu}{edge}\mspace{14mu}{points}} \\\left( {1 - Q_{2} - Q_{3}} \right)^{2} & {{if}\mspace{14mu} x_{k}\mspace{14mu}{is}\mspace{14mu}{one}\mspace{14mu}{of}\mspace{14mu} 36\mspace{14mu}{inner}\mspace{14mu}{points}}\end{matrix} \right.} & {(23)(c)}\end{matrix}$

which updates Equation (23)(b) to Equation (23)(d):

$\begin{matrix}\begin{matrix}{P_{c,N} = {P\left( {{\hat{x}}_{N} = x_{N}} \right)}} \\{= {{\frac{1}{16}\left( {1 - Q_{2}} \right)^{2}} + {\frac{3}{8}\left( {1 - Q_{2}} \right)\left( {1 - Q_{2} - Q_{3}} \right)} + {\frac{9}{16}\left( {1 - Q_{2} - Q_{3}} \right)^{2}}}}\end{matrix} & {(23)(d)}\end{matrix}$

Therefore, the sum of weighted spectral efficiencies for the (QPSK,16-QAM) GNC super-constellation is Equation (24):

$\begin{matrix}\begin{matrix}{S = {{w_{F}P_{c,F}} + {w_{N}P_{c,N}}}} \\{= {{w_{F}\left( {{\frac{3}{8}\left( {1 - Q_{1}} \right)} + {\frac{9}{16}\left( {1 - Q_{1}} \right)^{2}} + \frac{1}{16}} \right)} + {w_{N}\left( \left( {1 - Q_{2}} \right)^{2} \right)}}}\end{matrix} & (24)\end{matrix}$

where (w_(F), w_(N)) is a set of normalizing weighting coefficient as inEquation (5)(c). Alternatively, it could be updated by using Equation(22)(f) or (23)(d) Thus, the optimal α*_(F) for the (QPSK, 16-QAM) GNCsuper-constellation can be selected as Equation (25)(a):α*_(F)=arg max_(α) _(F) _(≥0.84)S  (25)(a)

If there is bit-swapping, α_(N) and α_(F) are switched, and so are theweighting coefficients. Accordingly, the optimal α⁻*_(F) for thebit-swapped GNC is given by Equation (25)(b):α⁻*_(F)=argmax_(α) _(F) ^(≤0.35)S  (25)(b)

D. Extension from Symbol to Codeword

As mentioned above, the idea of using weighted spectral efficienciescould be extended to a scheme using codeword level decoding. Morespecifically, Equation (26)(a) below could be used:S={tilde over (w)} _(N) P _(cw,{c,N}) +{tilde over (w)} _(F) P_(cw,{c,F})  (26)(a)

where P_(cw,{c,i}) is the codeword probability of being correctlydecoded for UE i. This could be empirically measured at a system level,or estimated by using a Mean Mutual Information per coded Bit (MMIB)mapping method. The MUST scheduler should consider several powerdistributions, MCSs, and RB allocations for each UE. The normalizedweighting coefficients could be changed to Equation (26)(b):

$\begin{matrix}{{\overset{\sim}{w}}_{i} = \frac{c_{i} + {\Delta_{i}\left( {c_{i},s_{i}} \right)}}{\Sigma_{k}\left( {c_{k} + {\Delta_{k}\left( {c_{k},s_{k}} \right)}} \right)}} & {(26)(b)}\end{matrix}$II. Bit-Swapping Rules when Using GNC

This section tabulates a decision rule for bit-swapping when usingsuperposition multiple access communication. Except for the (QPSK, QPSK)super-constellation, Gray mapping cannot be guaranteed for all powerdistribution sets. In those cases, bit swapping may be used to restoreGray mapping. However, this means a determination must be made attransmission about whether to bit-swap or not. In this section, a tableof decision parameters is generated according to an embodiment of thepresent invention. Below, the bit-swapping conditions are considered foreach super-constellation. A table at the end summarizes the results.

A. Bit-Swapping Conditions for (16-QAM, QPSK) GNC Super-Constellation

In a (16-QAM, QPSK) GNC super-constellation, such as FIG. 4, this firstcondition is needed to ensure Gray mapping:p(4−3q)≥−p(4−3q)  (27)(a)

which generates the first threshold: α_(F,1)=0.6429. The secondcondition is:p(4−q)≥−p(4−3q)  (27)(b)

which generates the second threshold: α_(F,2)=0.4444. The last thresholdis the bit-swapping threshold α_(F,3)=0.1667. These sets of thresholdsdetermine how much an individual constellation for each UE could beoverlapped in a joint/super constellation map for both UEs. In practice,performance could be affected by coding gains and bit-locations ofbit-loading. Because of this, α_(F,2) may be adjusted by Δ, i.e.,α_(F,2)+Δ (which will be seen in Table 2 at the end of this section).This, of course, applies to all super-constellations.

B. Bit-Swapping Conditions for (QPSK, 16-QAM) GNC Super-Constellation

In a (QPSK, 16-QAM) GNC super-constellation, such as FIG. 5, the firstGray mapping condition is:p(2−q)≥−p(2−q)  (28)(a)

which generates the first threshold: α_(F,1)=0.8333. The second Graymapping condition is:p(2−q)≥−p(6−q)  (28)(b)

which generates the second threshold: α_(F,2)=0.5556. The last thresholdis the bit-swapping threshold α_(F,3)=0.3571. These sets of thresholdsindicate the level of single constellation's overlap in a jointconstellation map.

C. Bit-Swapping Conditions for (16-QAM, 16-QAM) GNC Super-Constellation

For the (16-QAM, 16-QAM) GNC super-constellation, the feasible x valuescan be represented with p and q as p(12+3q), p(12+q), p(12−q), p(12−3q),p(4+3q), p(4+q), p(4−q), and p(4−3q) on the positive axis. Thus, Graymapping holds if the following condition is valid:−p(4−3q)≥p(4−3q)  (29)(a)

which generates the first threshold: α_(F,1)=0.9. As α_(F) decreases,some of the constellation points overlap each other. When the conditionindicated by Equation (29)(b) below is met, more than a half of thesymbol points in an individual constellation are overlapped with theother individual constellation for co-scheduled UEs in a joint/superconstellation map:−p(4−3q)≥p(12−3q)  (29)(b)

which generates the inequality for second threshold: α_(F,2)≥0.6923. Thelast threshold is the bit-swapping threshold α_(F,3)=0.1.

D. Bit-Swapping Conditions for (64-QAM, QPSK) GNC Super-Constellation

For the (64-QAM, QPSK) GNC super-constellation, the feasible x valuescan be represented with p and q as p(8+7q), p(8+5q), p(8+3q), p(8+q),p(8−q), p(8−3q), p(8−5q), and p(8−7q) on the positive axis. Thus, Graymapping holds if the following condition is valid:−p(8−7q)≥p(8−7q)  (30)(a)

which generates the first threshold: α_(F,1)=0.7. As α_(F) decreases,some of the constellation points overlap each other. Then, more than ahalf of the symbol points in an individual constellation will overlapwith the other individual constellation for co-scheduled UEs in ajoint/super constellation map if the conditions of Equations (30)(b) and(30)(c) are met:−p(8−7q)≥p(8−q)  (30)(b)−p(8−7q)≤p(8+q)  (30)(c)

which generates the second threshold: 0.3≤α_(F,2)≤0.4324. α_(F,2) isexpected to be in this range, with its exact position within the rangedepending on coding gains and bit locations. The bit-swapping thresholdis α_(F,3)=0.0455.

E. Bit-Swapping Conditions for (QPSK, 64-QAM) GNC Super-Constellation

For the (QPSK, 64-QAM) GNC super-constellation, the feasible x valuescan be represented with p and q as p(14+q), p(14−q), p(10+q), p(10−q),p(6+q), p(6−q), p(2+q), and p(2−q) on the positive axis. Thus, Graymapping holds if the following condition is valid:−p(2−q)≥p(2+q)  (31)(a)

which generates the first threshold: α_(F,1)=0.9545. As α_(F) decreases,some of the constellation points overlap each other. Then, more than ahalf of the symbol points in an individual constellation will overlapwith the other individual constellation for co-scheduled UEs in ajoint/super constellation map if Equation (31)(b) is met:−p(2−q)≥p(10−q)  (31)(b)

which generates the second threshold: α_(F,2)=0.7. The bit-swappingthreshold is α_(F,3)=0.3.

F. Summary of Bit-Swapping Conditions

The above results are summarized in Table 2 below:

TABLE 2 Decision Rule whether to Bit-Swap (Near, Far) Bit-Swapped(16QAM, QPSK) 0.6429 ≤ α_(F) ≤ 1.0000 OFF (Gray) 0.4444 + Δ ≤ α_(F) ≤0.6429 OFF 0.1667 ≤ α_(F) ≤ 0.4444 + Δ ON 0.0000 ≤ α_(F) ≤ 0.1667 ON(Gray) (QPSK, 16QAM) 0.8333 ≤ α_(F) ≤ 1.0000 OFF (Gray) 0.5556 + Δ ≤α_(F) ≤ 0.8333 OFF 0.3571 ≤ α_(F) ≤ 0.5556 + Δ ON 0.0000 ≤ α_(F) ≤0.3571 ON (Gray) (16QAM, 16QAM) 0.9000 ≤ α_(F) ≤ 1.0000 OFF (Gray)0.6923 + Δ ≤ α_(F) ≤ 0.9000 OFF 0.1000 ≤ α_(F) ≤ 0.6923 + Δ ON 0.0000 ≤α_(F) ≤ 0.1000 ON (Gray) (64QAM, QPSK) 0.7000 ≤ α_(F) ≤ 1.0000 OFF(Gray) 0.4324 + Δ ≤ α_(F) ≤ 0.7000 OFF 0.0455 ≤ α_(F) ≤ 0.4324 + Δ ON0.0000 ≤ α_(F) ≤ 0.0455 ON (Gray) (QPSK, 64QAM) 0.9545 ≤ α_(F) ≤ 1.0000OFF (Gray) 0.7000 + Δ ≤ α_(F) ≤ 0.9545 OFF 0.3000 ≤ α_(F) ≤ 0.7000 + ΔON 0.0000 ≤ α_(F) ≤ 0.3000 ON (Gray)

In the left column, the different super-constellation combinations arelisted, where (16-QAM, QPSK) means the near user is using 16-QAM and thefar user is using QPSK (making the super-constellation 64-QAM). In thecenter column, four different ranges are given for eachsuper-constellation, while the last column indicates what each range issuitable for, i.e., no bit-swapping (OFF), bit-swapping (ON), and whereGray encoding holds (Gray).

A series of simulations were performed, the results of which may be seenin U.S. Provisional Patent Application Ser. No. 62/210,326.

FIG. 6 is a flowchart of a more generic method of power allocationaccording to an embodiment of the present disclosure. In FIG. 6, asuperposition multiple access communication system capable of usinguniform and non-uniform superposition constellations(super-constellations) is assumed. In 610, for each receiver i receivingsuperposition multiple access transmission, calculating the conditionalprobability P_(c,i) of a bit being correctly received based on itslocation within the super-constellation. In 620, the normalizedweighting coefficient w_(i); is calculated for each receiver i. In 630,the sum S of weighted spectral efficiencies of all receivers i iscalculated using the conditional probability P_(c,i) and normalizedweighting coefficient W_(i) of each receiver i. In 640, the optimalpower allocation α*_(i) for receiver i is determined by maximizing thesum of weighted spectral efficiencies.

FIG. 7 is a flowchart of a method of power allocation according to anembodiment of the present disclosure. At 710, it is determined whetherthe superposition super-constellation as produced as a GNCsuper-constellation (“Indicate whether combined signals for MUST areproduced with Gray mapped constellation”). It is expected that the UEreceive high-layer signaling regarding the information ofsuper-positioned signals. At 720, the MCSs for the UEs are ready (“BothMCSs for two UEs are ready (even can be extended to multiple UEs,too)”). The UE could blindly estimate the modulation order by itself orreceive this information from eNB via high-layer signaling. If a codingrate of co-scheduled UE could not be blindly estimated, it is possibleto consider a high code rate for conservative operation. At 730, theprobability of being correct for UE i is derived where i=1, 2, 3, . . ., k, as explained herein. At 740, bias terms are loaded from the LUT toadjust effective MCSs. At 750, the sum of weighted spectral efficienciesis calculated. At 760, the power allocation for signals is distributedto the respective UEs.

Depending on the embodiment of the present disclosure, steps and/oroperations in accordance with the present disclosure may occur in adifferent order, or in parallel, or concurrently for different epochs,etc., in different embodiments, as would be understood by one ofordinary skill in the art. Similarly, as would be understood by one ofordinary skill in the art, FIGS. 6 and 7 are simplified representationsof the actions performed, and real-world implementations may perform theactions in a different order or by different ways or means. Similarly,as simplified representations, FIGS. 6 and 7 do not show other requiredsteps as these are known and understood by one of ordinary skill in theart and not pertinent and/or helpful to the present description.

Depending on the embodiment of the present disclosure, some or all ofthe steps and/or operations may be implemented or otherwise performed,at least in part, on a portable device. “Portable device” as used hereinrefers to any portable, mobile, or movable electronic device having thecapability of receiving wireless signals, including, but not limited to,multimedia players, communication devices, computing devices, navigatingdevices, etc. Thus, mobile devices include, but are not limited to,laptops, tablet computers, Portable Digital Assistants (PDAs), mp3players, handheld PCs, Instant Messaging Devices (IMD), cellulartelephones, Global Navigational Satellite System (GNSS) receivers,watches, cameras or any such device which can be worn and/or carried onone's person. “User Equipment” or “UE” as used herein corresponds to theusage of that term in the 3GPP LTE/LTE-A protocols, but is not in anyway limited by the 3GPP LTE/LTE-A protocols. Moreover, “User Equipment”or “UE” refers to any type of device, including portable devices, whichacts as a wireless receiver.

Depending on the embodiment of the present disclosure, some or all ofthe steps and/or operations may be implemented or otherwise performed,at least in part, using one or more processors running instruction(s),program(s), interactive data structure(s), client and/or servercomponents, where such instruction(s), program(s), interactive datastructure(s), client and/or server components are stored in one or morenon-transitory computer-readable media. The one or more non-transitorycomputer-readable media may be instantiated in software, firmware,hardware, and/or any combination thereof. Moreover, the functionality ofany “module” discussed herein may be implemented in software, firmware,hardware, and/or any combination thereof.

The one or more non-transitory computer-readable media and/or means forimplementing/performing one or more operations/steps/modules ofembodiments of the present disclosure may include, without limitation,application-specific integrated circuits (“ASICs”), standard integratedcircuits, controllers executing appropriate instructions (includingmicrocontrollers and/or embedded controllers), field-programmable gatearrays (“FPGAs”), complex programmable logic devices (“CPLDs”), and thelike. Some or all of any system components and/or data structures mayalso be stored as contents (e.g., as executable or other non-transitorymachine-readable software instructions or structured data) on anon-transitory computer-readable medium (e.g., as a hard disk; a memory;a computer network or cellular wireless network or other datatransmission medium; or a portable media article to be read by anappropriate drive or via an appropriate connection, such as a DVD orflash memory device) so as to enable or configure the computer-readablemedium and/or one or more associated computing systems or devices toexecute or otherwise use or provide the contents to perform at leastsome of the described techniques. Some or all of any system componentsand data structures may also be stored as data signals on a variety ofnon-transitory computer-readable transmission mediums, from which theyare read and then transmitted, including across wireless-based andwired/cable-based mediums, and may take a variety of forms (e.g., aspart of a single or multiplexed analog signal, or as multiple discretedigital packets or frames). Such computer program products may also takeother forms in other embodiments. Accordingly, embodiments of thisdisclosure may be practiced in any computer system configuration.

Thus, the term “non-transitory computer-readable medium” as used hereinrefers to any medium that comprises the actual performance of anoperation (such as hardware circuits), that comprises programs and/orhigher-level instructions to be provided to one or more processors forperformance/implementation (such as instructions stored in anon-transitory memory), and/or that comprises machine-level instructionsstored in, e.g., firmware or non-volatile memory. Non-transitorycomputer-readable media may take many forms, such as non-volatile andvolatile media, including but not limited to, a floppy disk, flexibledisk, hard disk, RAM, PROM, EPROM, FLASH-EPROM, EEPROM, any memory chipor cartridge, any magnetic tape, or any other magnetic medium from whicha computer instruction can be read; a CD-ROM, DVD, or any other opticalmedium from which a computer instruction can be read, or any othernon-transitory medium from which a computer instruction can be read.

While the invention has been shown and described with reference tocertain embodiments thereof, it will be understood by those skilled inthe art that various changes in form and detail may be made thereinwithout departing from the spirit and scope of the invention as definedby the appended claims.

What is claimed is:
 1. A method of power allocation in a superpositionmultiple access communication system capable of using uniform andnon-uniform superposition constellations (super-constellations),comprising: for each receiver i receiving superposition multiple accesstransmission, calculating the conditional probability P_(c,i) of asymbol being correctly received based on its location within asuper-constellation, wherein i is an index of integers from 1 to thetotal number of receivers receiving superposition multiple accesstransmission in the super-constellation; for the each receiver ireceiving superposition multiple access transmission, calculating anormalized weighting coefficient w_(i); calculating the sum S ofweighted spectral efficiencies of all of the each receiver i using thecalculated conditional probability P_(c,i) of the each receiver i andthe calculated normalized weighting coefficient w_(i) of the eachreceiver i; and determining the optimal power allocation α*_(i) for theeach receiver i by maximizing the sum of weighted spectral efficiencies.2. The method of claim 1, wherein the superposition multiple accesscommunication system uses Gray-mapped Non-uniform-capable Constellations(GNCs).
 3. The method of claim 1, wherein the superposition multipleaccess communication is Multi-User Superposition Transmission (MUST) ofthe Long Term Evolution (LTE) standard.
 4. The method of claim 1,wherein the conditional probability P_(c,i) is calculated using thefollowing equation:P _(c,i)=Σ_(k=1) ^(M) P({circumflex over (x)}_(k,i) =x _(k,i)), where{circumflex over (x)}_(k,i) denotes the detected symbol at the kthsymbol for receiver i.
 5. The method of claim 1, wherein the normalizedweighting coefficient w_(i) is calculated based on at least one of codegain, bit robustness relying on bit location, the Modulation and CodingScheme (MCS), and proportional fairness (PF).
 6. The method of claim 1,wherein the normalized weighting coefficient w_(i) is calculated usingthe following equation: $\begin{matrix}{{w_{i} = \frac{{c_{i}\log_{2}M_{i}} + {\Delta_{i}\left( {c_{i},s_{i}} \right)}}{\Sigma_{k}\left( {{c_{k}\log_{2}M_{k}} + {\Delta_{k}\left( {c_{k},s_{k}} \right)}} \right)}},} & \;\end{matrix}$ where C_(i) is the code rate for receiver i; S_(i) is aflag indicating whether receiver i's bits are swapped or not; andΔ_(i)(C_(i), s_(i)) is a bias term to compensate at least for the effectof coding gains between inner and outer bits, and is a function of C_(i)and S_(i).
 7. The method of claim 1, wherein the sum S of weightedspectral efficiencies of all receivers i is calculated using thefollowing equation: ${S = {\sum\limits_{i = 1}^{K}{w_{i}P_{c,i}}}},$where K is the total number of receivers, the probability P_(c,i) i of adetected symbol being correct is defined as:P _(c,i)=Σ_(k=1) ^(M) P({circumflex over (x)}_(k,i)=x_(k,i)), and{circumflex over (x)}_(k,i) denotes the detected symbol at the kthsymbol for receiver i.
 8. The method of claim 1, wherein there is only anear receiver and a far receiver and the sum S of weighted spectralefficiencies is calculated using the following equation:S=w _(F) P _(c,F) +w _(N) P _(c,N), where W_(F) is the weightedcoefficient for the far receiver, where W_(N)is the weighted coefficientfor the near receiver, the probability P_(c),_(i) of a detected symbolbeing correct is defined as:P _(c,i)=Σ_(k=1) ^(M) P({circumflex over (x)}_(k,i) =x _(k,i)), and{circumflex over (x)}_(k,i) denotes the detected symbol at the kthsymbol for receiver i.
 9. The method of claim 1, wherein the eachreceiver i comprises a near receiver and a far receiver; and determiningan optimal power allocation α*_(F) for the far receiver is calculated bymaximizing the sum of weighted spectral efficiencies using the followingequation:α*_(F)=argmax_(α) _(F) _(range)S, where α_(F) range is defined by themodulation orders of the near and far receivers, the number oftransmission layers, and whether there is bit-swapping, in the tablethat follows: “Far” “Near” receiver receiver Resulting “Super- α_(F)range α_(F) range constellation constellation constellation” Bit- forfor (2^(K) ^(F) )-QAM (2^(K) ^(N) )-QAM (2^(K) ^(F) ^(+K) ^(N) )-QAMSwapped Single Layer Two Layers QPSK QPSK 16-QAM OFF α_(F) ≥ 0.5 α_(F) ≥0.3333 ON α_(F) ≤ 0.5 α_(F) ≤ 0.3333 16-QAM QPSK 64-QAM OFF α_(F) ≥0.6429 α_(F) ≥ 0.4737 ON α_(F) ≤ 0.1667 α_(F) ≤ 0.0909 QPSK 16-QAM64-QAM OFF α_(F) ≥ 0.8333 α_(F) ≥ 0.7143 ON α_(F) ≤ 0.3571 α_(F) ≤0.2174 16-QAM 16-QAM 256-QAM OFF α_(F) ≥ 0.9 α_(F) ≥ 0.8182 ON α_(F) ≤0.1 α_(F) ≤ 0.0526 64-QAM QPSK 256-QAM OFF α_(F) ≥ 0.7 α_(F) ≥ 0.5385 ONα_(F) ≤ 0.0455 α_(F) ≤ 0.0233 QPSK 64-QAM 256-QAM OFF α_(F) ≥ 0.9545α_(F) ≥ 0.9130 ON α_(F) ≤ 0.3 α_(F) ≤ 0.1795.


10. The method of claim 1, wherein there is only a near receiver and afar receiver, the method further comprising: determining whether toperform bit-swapping between the near and far receivers, where thedetermination made depending on the value of α_(F) using the followingtable: (Near, Far) Bit-Swapped (16QAM, QPSK) 0.6429 ≤ α_(F) ≤ 1.0000 OFF(Gray) 0.4444 + Δ ≤ α_(F) ≤ 0.6429 OFF 0.1667 ≤ α_(F) ≤ 0.4444 + Δ ON0.0000 ≤ α_(F) ≤ 0.1667 ON (Gray) (QPSK, 16QAM) 0.8333 ≤ α_(F) ≤ 1.0000OFF (Gray) 0.5556 + Δ ≤ α_(F) ≤ 0.8333 OFF 0.3571 ≤ α_(F) ≤ 0.5556 + ΔON 0.0000 ≤ α_(F) ≤ 0.3571 ON (Gray) (16QAM, 16QAM) 0.9000 ≤ α_(F) ≤1.0000 OFF (Gray) 0.6923 + Δ ≤ α_(F) ≤ 0.9000 OFF 0.1000 ≤ α_(F) ≤0.6923 + Δ ON 0.0000 ≤ α_(F) ≤ 0.1000 ON (Gray) (64QAM, QPSK) 0.7000 ≤α_(F) ≤ 1.0000 OFF (Gray) 0.4324 + Δ ≤ α_(F) ≤ 0.7000 OFF 0.0455 ≤ α_(F)≤ 0.4324 + Δ ON 0.0000 ≤ α_(F) ≤ 0.0455 ON (Gray) (QPSK, 64QAM) 0.9545 ≤α_(F) ≤ 1.0000 OFF (Gray) 0.7000 + Δ ≤ α_(F) ≤ 0.9545 OFF 0.3000 ≤ α_(F)≤ 0.7000 + Δ ON 0.0000 ≤ α_(F) ≤ 0.3000 ON (Gray).


11. An apparatus for power allocation in a superposition multiple accesscommunication system capable of using uniform and non-uniformsuperposition constellations (super-constellations), comprising: atleast one non-transitory computer-readable medium storing instructionscapable of execution by a processor; and at least one processor capableof executing instructions stored on the at least one non-transitorycomputer-readable medium, where the execution of the instructionsresults in the apparatus performing a method comprising: for eachreceiver i receiving superposition multiple access transmission,calculating the conditional probability P_(c,i) of a symbol beingcorrectly received based on its location within a super-constellation,wherein i is an index of integers from 1 to the total number ofreceivers receiving superposition multiple access transmission in thesuper-constellation; for the each receiver i receiving superpositionmultiple access transmission, calculating a normalized weightingcoefficient w_(i); calculating the sum S of weighted spectralefficiencies of all of the each receiver i using the calculatedconditional probability P_(c,i) of the each receiver i and thecalculated normalized weighting coefficient w_(i) of the each receiveri; and determining the optimal power allocation α*_(i); for the eachreceiver i by maximizing the sum of weighted spectral efficiencies. 12.The apparatus of claim 11, wherein the superposition multiple accesscommunication system uses Gray-mapped Non-uniform-capable Constellations(GNCs).
 13. The apparatus of claim 11, wherein the superpositionmultiple access communication is Multi-User Superposition Transmission(MUST) of the Long Term Evolution (LTE) standard.
 14. The apparatus ofclaim 11, wherein the normalized weighting coefficient w_(i) iscalculated based on at least one of code gain, bit robustness relying onbit location, the Modulation and Coding Scheme (MCS), and proportionalfairness (PF).
 15. The apparatus of claim 11, wherein the normalizedweighting coefficient w_(i) is calculated using the following equation:${w_{i} = \frac{{c_{i}\log_{2}M_{i}} + {\Delta_{i}\left( {c_{i},s_{i}} \right)}}{\Sigma_{k}\left( {{c_{k}\log_{2}M_{k}} + {\Delta_{k}\left( {c_{k},s_{k}} \right)}} \right)}},$where C_(i) is the code rate for receiver i; S_(i) is a flag indicatingwhether receiver i's bits are swapped or not; and Δ_(i) (C_(i), S_(i))is a bias term to compensate at least for the effect of coding gainsbetween inner and outer bits, and is a function of C_(i) and S_(i) . 16.The apparatus of claim 11, wherein the sum S of weighted spectralefficiencies of all receivers i is calculated using the followingequation: ${S = {\sum\limits_{i = 1}^{K}{w_{i}P_{c,i}}}},$ where K isthe total number of receivers, the probability P_(c,i) of a detectedsymbol being correct is defined as:P _(c,i)=Σ_(k=1) ^(M) P({circumflex over (x)}_(k,i) =x _(k,i)), and{circumflex over (x)}_(k,i) denotes the detected symbol at the kthsymbol for receiver i.
 17. The apparatus of claim 11, wherein there isonly a near receiver and a far receiver and the sum S of weightedspectral efficiencies is calculated using the following equation:S=w _(F) P _(c,F) +w _(N) P _(c,N), where w_(F) is the weightedcoefficient for the far receiver, where w_(N) is the weightedcoefficient for the near receiver, the probability P_(c,i) of a detectedsymbol being correct is defined as:P _(c,i)=Σ_(k=1) ^(M) P({circumflex over (x)}_(k,i) =x _(k,i)), and{circumflex over (x)}_(k,i) denotes the detected symbol at the kthsymbol for receiver i.
 18. The apparatus of claim 11, wherein the eachreceiver i comprises a near receiver and a far receiver; and determiningan optimal power allocation α*_(F) for the far receiver is calculated bymaximizing the sum of weighted spectral efficiencies using the followingequation:α*_(F)=argmax₆₀ _(F) _(range)S, where α_(F) range is defined by themodulation orders of the near and far receivers, the number oftransmission layers, and whether there is bit-swapping, in the tablethat follows: “Far” “Near” receiver receiver Resulting “Super- α_(F)range α_(F) range constellation constellation constellation” Bit- forfor (2^(K) ^(F) )-QAM (2^(K) ^(N) )-QAM (2^(K) ^(F) ^(+K) ^(N) )-QAMSwapped Single Layer Two Layers QPSK QPSK 16-QAM OFF α_(F) ≥ 0.5 α_(F) ≥0.3333 ON α_(F) ≤ 0.5 α_(F) ≤ 0.3333 16-QAM QPSK 64-QAM OFF α_(F) ≥0.6429 α_(F) ≥ 0.4737 ON α_(F) ≤ 0.1667 α_(F) ≤ 0.0909 QPSK 16-QAM64-QAM OFF α_(F) ≥ 0.8333 α_(F) ≥ 0.7143 ON α_(F) ≤ 0.3571 α_(F) ≤0.2174 16-QAM 16-QAM 256-QAM OFF α_(F) ≥ 0.9 α_(F) ≥ 0.8182 ON α_(F) ≤0.1 α_(F) ≤ 0.0526 64-QAM QPSK 256-QAM OFF α_(F) ≥ 0.7 α_(F) ≥ 0.5385 ONα_(F) ≤ 0.0455 α_(F) ≤ 0.0233 QPSK 64-QAM 256-QAM OFF α_(F) ≥ 0.9545α_(F) ≥ 0.9130 ON α_(F) ≤ 0.3 α_(F) ≤ 0.1795.


19. The apparatus of claim 11, wherein there is only a near receiver anda far receiver, where the method performed by the apparatus furthercomprises: determining whether to perform bit-swapping between the nearand far receivers, where the determination made depending on the valueof α_(F) using the following table: (Near, Far) Bit-Swapped (16QAM,QPSK) 0.6429 ≤ α_(F) ≤ 1.0000 OFF (Gray) 0.4444 + Δ ≤ α_(F) ≤ 0.6429 OFF0.1667 ≤ α_(F) ≤ 0.4444 + Δ ON 0.0000 ≤ α_(F) ≤ 0.1667 ON (Gray) (QPSK,16QAM) 0.8333 ≤ α_(F) ≤ 1.0000 OFF (Gray) 0.5556 + Δ ≤ α_(F) ≤ 0.8333OFF 0.3571 ≤ α_(F) ≤ 0.5556 + Δ ON 0.0000 ≤ α_(F) ≤ 0.3571 ON (Gray)(16QAM, 16QAM) 0.9000 ≤ α_(F) ≤ 1.0000 OFF (Gray) 0.6923 + Δ ≤ α_(F) ≤0.9000 OFF 0.1000 ≤ α_(F) ≤ 0.6923 + Δ ON 0.0000 ≤ α_(F) ≤ 0.1000 ON(Gray) (64QAM, QPSK) 0.7000 ≤ α_(F) ≤ 1.0000 OFF (Gray) 0.4324 + Δ ≤α_(F) ≤ 0.7000 OFF 0.0455 ≤ α_(F) ≤ 0.4324 + Δ ON 0.0000 ≤ α_(F) ≤0.0455 ON (Gray) (QPSK, 64QAM) 0.9545 ≤ α_(F) ≤ 1.0000 OFF (Gray)0.7000 + Δ ≤ α_(F) ≤ 0.9545 OFF 0.3000 ≤ α_(F) ≤ 0.7000 + Δ ON 0.0000 ≤α_(F) ≤ 0.3000 ON (Gray).